Let's determine the equations of the parabolas shown in the image by analyzing their vertex form. The general equation of a parabola in vertex form is:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola and $a$ determines whether the parabola opens upwards or downwards (as well as how wide or narrow it is).

a) Parabola $a$

• Vertex: $(1, 6)$
• The parabola opens downwards (narrow).

The equation for this parabola is: $y = -a(x - 1)^2 + 6$ To find $a$, we need an additional point, but since the graph doesn't provide one explicitly, we can leave $a$ undefined or assume it is 1, making the equation: $y = -(x - 1)^2 + 6$

b) Parabola $b$

• Vertex: $(1, 5)$
• The parabola opens upwards.

The equation is: $y = a(x - 1)^2 + 5$ Assuming $a = 1$, the equation becomes: $y = (x - 1)^2 + 5$

c) Parabola $c$

• Vertex: $(3, 9)$
• The parabola opens upwards.

The equation is: $y = a(x - 3)^2 + 9$ Assuming $a = 1$: $y = (x - 3)^2 + 9$

d) Parabola $d$

• Vertex: $(0, -4)$
• The parabola opens downwards.

The equation is: $y = a(x - 0)^2 - 4 \quad \text{or simply} \quad y = a x^2 - 4$ Assuming $a = 1$: $y = -x^2 - 4$

e) Parabola $e$

• Vertex: $(1, \frac{1}{2})$
• The parabola opens upwards.

The equation is: $y = a(x - 1)^2 + \frac{1}{2}$ Assuming $a = 1$: $y = (x - 1)^2 + \frac{1}{2}$

f) Parabola $f$

• Vertex: $(1, 1)$
• The parabola opens downwards.

The equation is: $y = a(x - 1)^2 + 1$ Assuming $a = -1$: $y = -(x - 1)^2 + 1$

Would you like to dive deeper into any of these parabolas or see more details? Here are five follow-up questions to consider:

1. How do you find the value of $a$ using another point on the parabola?
2. What is the significance of the vertex in the parabola's graph?
3. How can you determine if a parabola opens upwards or downwards from the equation?
4. How does changing the value of $a$ affect the width of the parabola?
5. Can you convert these equations to standard form?

Tip: The sign of $a$ in the vertex form of a parabola ($y = a(x - h)^2 + k$) dictates whether the parabola opens upwards (positive $a$) or downwards (negative $a$).